Piezo robotic hand for motion manipulation from micro to macro

Multiple degrees of freedom (DOFs) motion manipulation of various objects is a crucial skill for robotic systems, which relies on various robotic hands. However, traditional robotic hands suffer from problems of low manipulation accuracy, poor electromagnetic compatibility and complex system due to limitations in structures, principles and transmissions. Here we present a direct-drive rigid piezo robotic hand (PRH) constructed on functional piezoelectric ceramic. Our PRH holds four piezo fingers and twelve motion DOFs. It achieves high adaptability motion manipulation of ten objects employing pre-planned functionalized hand gestures, manipulating plates to achieve 2L (linear) and 1R (rotary) motions, cylindrical objects to generate 1L and 1R motions and spherical objects to produce 3R motions. It holds promising prospects in constructing multi-DOF ultra-precision manipulation devices, and an integrated system of our PRH is developed to implement several applications. This work provides a new direction to develop robotic hand for multi-DOF motion manipulation from micro scale to macro scale.

upper and lower surfaces (see blue arrow), it will produce extensional or contractional mechanical deformations (see magenta arrow) along the thickness direction, and the deformation direction depends on the electric field. This is also known as d33 working mode of the piezoelectric ceramic. As for the actuation part of the piezo finger, it can be regarded as a cylindrical body with piezoelectric characteristics and partitioned cross section ( Supplementary   Fig. 2e); according to the above structural and electrical configurations, the actuation part can adopt zonal polarization and excitation strategies to generate bending and extending motions as follows: (i) when the same exciting electrical field is applied to the partitions of 1 and 3 or the partitions of 2 and 4, the actuation part produces bending motion because that one side is compressed and the other side is extended; (ii) when one exciting electrical field is applied to the partitions of 1 and 2, and its reverting electrical field is applied to the partitions of 3 and 4 simultaneously, the actuation part can generate extending or contracting motion along the vertical direction ( Supplementary Fig. 2f). To sum up, the actuation part can produce multidimensional bending and extending motions by accumulating basic deformations of ten groups of actuation units, this can be further used to induce the multi-dimensional motions of the whole piezo finger.
In fact, we can also use other polarization configurations and schemes to realize similar motions. A representative configuration is shown in Supplementary Fig. 2h to Fig. 2j. In detail, as shown in Supplementary Fig. 2h, two groups of piezo rings can be configurated to form an actuation part (divided as top half and bottom half), in which the top half and the bottom half contain several actuation unit-I and several actuation unit-II, respectively. The copper electrode slices are set between the adjacent two piezo rings to apply electric fields on them. It should be noted that every piezo ring in unit-I and unit-II holds two polarized regions, and the piezo rings in unit-I and that in unit-II are configurated to mutually orthogonal ( Supplementary Fig. 2i).
With these configurations, the top half and the bottom half can separately produce 1-dimensional bending deformation when the same electric fields are applied on them simultaneously, while they can also produce 1-dimentisonal extending deformation when the opposite electric fields are applied on them simultaneously. Therefore, the whole actuation part integrating piezo rings with two polarized regions can produce bending deformations along x and y axes, and extending deformation along z axis ( Supplementary Fig. 2j). It should be noted that this configuration will lead to response difference in the two bending deformations. This is because that there is an amplification effect (by the height of the top half of the actuation part) on the response when the bottom half of the actuation part is stimulated (assuming to excite bending deformation along x axis), whereas there is no amplification effect when the top half is stimulated (assuming to excite bending deformation along y axis). Therefore, in order to avoid the response difference in the two lateral bending motions of a piezo finger when using the same number of piezo rings to stimulate motions, the configurations of using piezo rings with four polarized regions are utilized in this work.

Material parameters of the PRH components
In terms of material configurations of the PRH components, the palm and the finger bases are stainless steel with good rust resistance (Density of 7.75×10 3 kg/m 3 , Young's modulus of 1.93×10 11 N/m 2 , and Poisson's ratio of 0.31); the common electrodes and the fan-shaped electrodes are set as beryllium bronze with good electrical conductivity (Density of 8.
where d, ε T and c E are the piezoelectric constant matrix, the dielectric matrix at constant stress and the stiffness matrix at constant electric field, respectively.

Supplementary Note 4
The influence of changing the number of the piezoelectric ceramic rings on the response displacement and response speed of the piezo finger.
We also analyzed the influence of the number of the piezo rings on the output displacement of the piezo finger by finite-element-method simulation with ANSYS. Meanwhile, the influence of the exciting voltage on the output displacement with different number of piezo rings is also simulated. Take the lateral bending motion as an example, the output displacement of the piezo finger is approximately linear with the number of the piezo rings (Fig. 2g), which means that the output displacement can be adjusted to meet some potential demands by changing the number of piezo rings. It should be noted that the simulation results in two bending motion directions are same due to structural symmetry. Quantitatively, the simulated lateral output displacement of the piezo finger is about 24 μm under the maximum exciting voltage of 600 Vp-p when the number of the piezo rings are set as 20 PCS. When the number of the piezo rings increases to 2 times the situation used in this work (i.e., 40 PCS), the output displacement ascends to about 58.6 μm (relative to 2.44 times that under piezo rings of 20 PCS).
The influence of the number of piezo rings on the response speed can be analyzed qualitatively from the electrical-mechanical conversion processes. Theoretically, piezo rings are typical capacitance components, and they can produce response by two processes: (i) the exciting voltage from the external excitation devices is applied to the piezo rings to charge them; (ii) the piezo rings produce conversion of electrical energy to mechanical energy for producing deformations. The piezo rings theoretically need charging time t1 and conversion time t2 to complete these two processes, respectively. It should be noted that the charging time t1 is more dominant than time t2 due to the fast conversion characteristic of the piezoelectric ceramic, which means that the response time of the piezo rings mainly depends on the charge time t1. As for the charge time t1 of piezo rings, it mainly depends on the capacitance level of the piezo rings (determining how much energy needs to be charged) and the output current of the external excitation device (determining how fast to charge the energy). Single piezo ring is actually a parallel plate capacitance structure, and there is an electrical parallel relationship in multiple piezo rings. Therefore, the total capacitance will change when the number of the piezo ring changes, which affects the charge time t1 and further affects the response speed of the piezo finger.
The capacitance level of the piezo rings can be described with mathematical expression as follows: where, Cp is the total capacitance of one group of piezo rings used to generate 1-dimensioanl motion; n is the number of the piezo rings; S shows the relative electrode area of each piezo ring; d is the thickness of each piezo ring;  is permittivity of piezo ceramic; it should be noted that the piezo rings are assumed to hold the same size (area Si and thickness di) in the above equation.
The charging time of the piezo rings can be approximately expressed as follows: where, U and I are the exciting voltage applied on the piezo rings and output current of the external excitation device. The above equations indicate that the charging time t1 is proportion to the number of piezo rings in theory. The total response time t can be represented as: The conversion time t2 can be ignored as it is short enough for piezoelectric ceramic, then the response speed can be regarded as only relation to charging time t1. In another word, under the same external excitation conditions, changing the number of piezo rings with n times will cause an approximately n times change in response time.
In general, the average response speed can be regarded as the ratio between the output displacement and the response time. It should be noted that the increase of the number of the piezo rings n not only leads to the increase of the output displacement, but also causes the increase of the response time. Therefore, the influence of increasing the number of the piezo rings n on the response speed can be analyzed by estimating the increase level of the output displacement and the response time. (i) According to the above equation about response time t, if the number of the piezo rings n increase to 2n, the response time will increase to 2 times that corresponding to the number of n. (ii) According to the simulation results shown in Fig. 2g, when the number of the piezo rings n=20 increases to 2n=40, the output displacement will increase to about 2.44 times of that corresponding to n=20. These results show that the change level of the output displacement is more than that of the response time when changing the number of piezo rings, which means that the increase of the number of piezo rings leads to the increase of the response speed in theory.

Characteristics of the piezo fingers
The fundamental characteristics of the piezo fingers just reflect the characteristics of the whole PRH, which determine the final manipulation performance. Therefore, we implemented several experiments to investigate the fundamental characteristics of the piezo fingers.
We investigated the input-output linearity of the four piezo fingers. The lateral bending motions of each piezo finger were excited by applying a sinusoidal signal with voltage of 600 Vpp and frequency of 1 Hz, and the responding lateral displacements on the top of the fingertips were measured. Then the input-output characteristics, namely the relationships between the exciting voltages and the output displacements, were obtained ( Supplementary Fig. 4a). The results show that the output displacements are linear with the exciting voltages, and there are slight nonlinear hysteresis effects caused by the inherent nonlinear characteristics of the piezoelectric materials ( Supplementary Fig. 4b). Quantitatively, as for the four piezo fingers, the nonlinear hysteresis ratios of the lateral output displacements are within 3.95%, which is benefited from fast electrical and mechanical responses of piezoelectric ceramic. This feature helps the PRH to achieve high precision dynamic motion manipulation.
In order to investigate the displacement resolution (minimum achievable displacement) of the piezo fingers, we set the exciting signal as a stepping signal with ten equal increments and ten equal decrements during one second. We set the voltage increment and decrement as several volts and tested the displacement to see if it could produce stable displacement increments and decrements. By gradually reducing the voltage change to one step of 0.5 V, the displacement increments and decrements were still stable ( Supplementary Fig. 4c). Then the average of the ten increments was calculated to evaluate displacement resolution of the piezo fingers ( Supplementary Fig. 4d). These results indicate that the displacement resolutions of the four piezo fingers are within 15.45 nm.
In view of that a significant advantage of piezoelectric ceramic is fast response, the piezo finger constructed on this material can inherit fast response ability resultantly. We chose the bending motions of one finger as an example to evaluate fast response ability. The saw-tooth signals with voltage of 600 Vp-p, frequency of 1 Hz, symmetry of 0% and 100% were applied to stimulate lateral motions of the piezo finger in two directions, respectively. Then the response displacements were obtained ( Supplementary Fig. 4e). These results indicate that the response time to produce maximum displacement of more than 20 μm is within 0.5 ms. According to the classical kinematic law, we can estimate that the response acceleration reaches about 160 m/s 2 .
A Doppler laser vibration testing system was used to acquire the vibration characteristics of the piezo fingers ( Supplementary Fig. 4f). The tested results show that the first-order natural frequencies of the four piezo fingers are 4.06 kHz, 4.09 kHz, 4.06 kHz and 4.10 kHz, respectively. These results fully reveal that the piezo fingers hold features of high natural frequency and high stiffness characteristics, which help the PRH to adapt requirements of large carrying load and high manipulation frequency.
To sum up, for the lateral bending motions of the piezo fingers, we find that they hold the significant characteristics including multi-dimensional motion, low hysteresis, high resolution, fast response and high natural frequency. Thus, the overall PRH holds these characteristics resultantly due to the configuration integration of four piezo fingers. These fundamental characteristics obviously surpass those of the other robotic hands, which can bring many advantages in the motion manipulations.

Supplementary Note 6 Functionalized hand gestures using longitudinal motion of the piezo fingers
We  Fig. 5f), which are just the reverse situations of gesture 12. The abovementioned functionalized hand gestures will bring more flexible motion manipulation ability of our PRH.

Supplementary Note 7 Description for noise experiments of the PRH
Good man-machine compatibility is a factor worth considering, but the working frequency of the PRH at hundreds of Hertz may produce working noise due to the fast excitation of the piezoelectric ceramic components. Thus, we carried out noise experiment of the PRH under maximum working voltage of 600 Vp-p and different working frequencies, in which a digital noise meter is used to capture the noise level of the PRH when working (Supplementary Fig. 8c).
The tested results show that the noise level of the PRH is less than 53.74 dB under maximum working frequency of 270 Hz and less than 57.42 dB under the maximum tested frequency of 360 Hz (Supplementary Fig. 4g). The working noise 53.74 dB under the maximum working frequency of 270 Hz is within acceptable level of human ear, just liking the noise level of loud talking (far away from the general endurable limit 100 dB of human ear). It is worth noting that the potential working scenario of our PRH is intermittent high-precision motion manipulation, and its working frequency can be set as low level, which enables us to maintain the noise level within the comfortable range.

Supplementary Note 8 Description for thermal characteristic experiments
In order to evaluate the heating level of the PRH when working, we carried out a thermal  Fig. 4h and Fig. 4i). The thermal images of the PRH under different working time are shown in Supplementary Fig. 4j to Fig. 4m. These experiments reflect that our PRH has no heating phenomenon when working continuously with the maximum working voltage and frequency. This helps to ensure the stability of its own characteristics and avoid the influence of the PRH on the manipulated objects. The temperature around the piezo finger is unchanged when working, which means that the response characteristics including the response speed and amplitude of the piezo finger cannot be affected by the nonexistent heating problem. Thus, the operated samples of the PRH are also not affected by the heating problem.

Supplementary Note 9 Characteristic comparison between the plate manipulated with the PRH and other precision stages
Manipulating the plate to produce two translational DOFs and one rotary DOF motions is an important capability of our PRH, which is experimentally evaluated in detail in this work. In order to further evaluate the level of its manipulation characteristics, a simple characteristic comparison between the motion plate manipulated with our PRH and other precision stages is accomplished (Supplementary Table 3). The compared items contain the motion DOF, principle, overall size, motion stroke, working voltage, working frequency and load capability. The compared results show that the motion plate manipulated with our PRH holds several merits: (i) the motion plate manipulated with our PRH achieves greater motion strokes; (ii) the working frequency is more than other motion stages; (iii) the load capability of the motion plate manipulated with our PRH achieves excellent capability to carry other objects for motions. It is worth noting that the plate manipulated with our PRH is only a construction case of many promising functions. In the follow-up work, we will also consider using the PRH to build multi-DOF devices for specific applications.

Experimental system and measurement method
We used an xPC system ( Supplementary Fig. 8a)  where θ is the rotary displacement of the square plate around Zp axis; x1 and x2 denote the measured displacements acquired by the two laser heads; Lp is the distance between the two laser spots emitted by two laser heads. Note: this indirect method to obtain rotary displacement is only suitable for the small rotary angle.